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0/0 form Evaluate the following limits using l'Hôpital's Rule. $$\lim _{x \rightarrow 1} \frac{\ln x}{4 x-x^{2}-3}$$

   0/0 form Evaluate the following limits using l'Hôpital's Rule.
$$\lim _{x \rightarrow 1} \frac{\ln x}{4 x-x^{2}-3}$$

Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle… 1st Edition

Chapter 4, Problem 15

Instant Answer

Solved by Expert Mike Gaerlan

05/31/2020

Step 1

So, we differentiate the numerator and the denominator separately. The derivative of $\ln x$ is $1/x$ and the derivative of $4x - x^2 - 3$ is $4 - 2x$. So, we have: $$\lim _{x \rightarrow 1} \frac{1/x}{4 - 2x}$$ Show more…

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0/0 form Evaluate the following limits using l'Hôpital's Rule. $$\lim _{x \rightarrow 1} \frac{\ln x}{4 x-x^{2}-3}$$

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Key Concepts

Differentiation

The process of computing the derivative of a function, which represents how the function's value changes at any given point. In the context of limit evaluation with l'Hôpital's Rule, differentiation is used to transform indeterminate forms into limits that can be easily evaluated.

l'Hôpital's Rule

A mathematical technique used to evaluate limits that initially result in indeterminate forms, such as 0/0 or ?/?. The rule states that if the limit of a ratio of two functions yields an indeterminate form, the limit can be computed by taking the derivative of the numerator and the derivative of the denominator, provided that the new limit exists.

Indeterminate Forms

Expressions where direct substitution in a limit does not yield a clear answer, typically resulting in forms like 0/0 or ?/?. These forms signal that further analysis or manipulation, such as applying l'Hôpital's Rule, is needed to determine the limit.

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